Thales’ Theorem and Lockhart’s Lament...

The Yorkshire Branch of the Mathematical Association recently hosted a talk by David Acheson entitled Proof, Pizza and Guitar. (By the way I’ll be giving a talk on card cheating for the YBMA on Wednesday 8th February at 7.30pm in School of Mathematics. All welcome but a small charge of a pound may be made.) During David’s talk he gave a proof of Thales’ Theorem. This is a theorem that states the following. For any point in a semi-circle, the angle formed by the lines from that point to the two edge points of the base is right-angled. This is a good theorem in that, to me at least, it does not seem intuitively obvious (what, it’s always a right angle? Really?) and yet it is easy to convince yourself it’s true by doing some examples. Thales (c624BC – c547BC) is often considered to be the first scientist because he was the first person (we know of) who looked for non-supernatural reasons for phenomena. Rather than believing lightning or earthquakes were caused by gods he considered more natural explanations. However, his solution to the latter involved the way that the land floated on the sea, i.e., he was totally wrong but here it is the concept of avoiding invoking the gods that counts. In the case of mathematics he is credited with a number of theorems and the main point is that, allegedly, he provided proof. He is also credited with measuring the pyramids in Egypt. His method is interesting because it does not involve a brute force use of measuring instruments, i.e., get out measuring rods and send people up the pyramids with them. His proof is more elegant than that. He measured the height of a slave and when the sun was...

Four strategies for better study...

In Cal Newport’s Study Hacks blog he recently had a post about a piano student called Jeremy. Jeremy’s Strategies for Becoming Excellent… Strategy #1: Avoid Flow. Do What Does Not Come Easy. “The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.” Strategy #2: To Master a Skill, Master Something Harder. “Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.” Strategy #3: Systematically Eliminate Weakness. “Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.” Strategy #4: Create Beauty, Don’t Avoid Ugliness. “Weak pianists make music a reactive task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.” Of course as we are talking about studying for a public performance of music these strategies don’t translate immediately or perfectly to...

Paper review – with maths!...

I was back on the radio reviewing the morning papers. Fortunately, there was a lot of good maths and science stories. Unfortunately, too many to cover. You can hear me (for the next few days at least) at BBC Radio Leeds at approximately 1:25 and 1:55. The main stories were Alex Bellos: How to Learn to Love Maths in the Guardian, an interview with Patrick Moore in the Daily Mail and an astronomy article in the Daily...

Maths Faculty Podcasts...

Back in the summer I, along with a number of other colleagues, recorded some video podcasts for themathsfaculty.org. These videos are designed for A-Level students and I gave two lectures: one on induction and one on trigonometric identities. (I was supposed to give a third on the Millenium Prizes but didn’t due to a back injury. The material I created for this will appear elsewhere – stay tuned!). The recorded videos can now be seen on the Maths Faculty website: Here’s induction and here’s trig identities. Look out for the great domino toppling sequence at about 1:56 in the induction video. We did have problems recording as we did not have any tele-prompters and so I was always glancing at my notes rather than looking at the camera. If there is a next time I might just give the lecture without referring to notes. Makes it more natural. Any feedback would be appreciated....